We have thus successfully provided boundary conditions at spatial infinity for supergravity that are invariant under an extension of the BMS\(_4\)  algebra by an infinite number of fermionic generators. These can be thought of as square roots of the supertranslations. One central idea in the construction is to enlarge the boundary conditions of [1]}, which lead to a finite-dimensional fermionic extension of the BMS\(_4\)  algebra, by allowing an improper gauge term \(\partial _k \chi \)  that decays slowlier at infinity in the gravitino field \(\psi _k\) .
